Saturday, 7 July 2018

sequences and series - The Lerch transcendent evaluation for the parameters HurwitzLerchPhi[z,-4 s,0]



I want to evaluate the Hurwitz zeta function
Φ(z,4s,0)=k=0zkk4s


And |z|<1 and s>1.
I want to have un upper bound for it.
I tried even Wolfram Mathematica (to have some hint of the form if possible fot the calculus) , but without success (since I give parameters and no numbers as an input).


Answer



The OP is asking about the upper bound of the function:




Φ(z,4s,0)=k=1k4szk



|z|<1s>1



First, it's obvious that:



k=1k4szkk=1k4s|z|k



So let us consider only z>0.




For a fixed z we can see that:



p>qk=1k4pzk>k=1k4qzk



Which means that if an upper bound exists for Φ as a function of s, it will be the limit:



lims+k=1k4szk=







Now let us fix a finite s and see what happens for z1. For z=1 the series obviously diverges, which automatically means that for z close to 1 the value can be as large as we want, which means there's no upper bound for a fixed s either.



More rigorously, we need to prove that for any N>0 there exists ϵ>0 such that:



k=1k4s(1ϵ)k>N



It's rather easy, we can just compare to geometric series:



k=1k4s(1ϵ)k>k=1(1ϵ)k=1ϵϵ=1ϵ1




Now pick ϵ=1N+1 and the proof is finished.


No comments:

Post a Comment

real analysis - How to find limhrightarrow0fracsin(ha)h

How to find limh0sin(ha)h without lhopital rule? I know when I use lhopital I easy get $$ \lim_{h\rightarrow 0}...